Quasi-two-body $B_{(s)}\to V \pi\pi$ Decays with Resonance $f_0(980)$ in PQCD Approach

Motivated by the measurements of branching fractions of the quasi-two-body decays $B^0\to K^{*0}(f_0(980)\to )\pi^+\pi^-$ and $B^0\to \rho^0(f_0(980)\to )\pi^+\pi^-$, we study the charmless $B\to V (f_0(980)\to ) \pi^+\pi^-$ decays in the perturbative QCD approach. Supposing that $f_0(980)$ is a two-quark state and mixture of $n\bar n=(u\bar u+d\bar d)/\sqrt 2$ and $s\bar s$ with the mixing angle $\theta$, we calculate the branching fractions of these decays with the new introduced $s$-wave $\pi^+\pi^-$-pair wave function. When the mixing angle $\theta$ lies in the range $[135^{\circ}, 155^{\circ}]$, the calculated branching fractions of the $B^0\to K^{*0}f_0(980)\to K^{*0}\pi^+\pi^-$ and $B^0\to \rho^0f_0(980)\to \rho^0\pi^+\pi^-$ decays are in agreement with the experimental data. The branching fractions of other decays could be measured in the current LHCb and Belle II experiments. Considering the isospin symmetry, we also estimate the branching fractions of the quasi-two-body decays $B\to Vf_0(980)\to V\pi^0\pi^0$, which are half of those the corresponding decays $B\to Vf_0(980)\to V\pi^+\pi^-$. Moreover, the direct $CP$ asymmetries of these decays are also calculated, and some of them can be tested in the current experiments.

Unlike the two-body decays where the kinematics is fixed, three-body decay amplitudes depend on two kinematic variables. For a decay B(p B ) → M 1 (p 1 )M 2 (p 2 )M 3 (p 3 ), it is general to define the variables as two invariant masses of two pairs of final state particles, for instance, s 12 and s 13 with the definition s i j = 2(p i · p j )/m 2 B . All physical kinematics configurations could define a two-dimensional region in the s 12 − s 13 plane, and the density plot of the differential decay rate dΓ/ds 12 ds 13 in this region is called a Dalitz plot. Specially, when the final states are the light mesons such as the π and K mesons, the corresponding configuration reduces to a triangle region. In general, the Dalitz plot has three typical regions according to the characteristic kinematics. The central region so-called "Mercedes Star" configuration corresponds to the case where all the invariant masses are roughly the same and of order of m B . In this region all three light mesons have a large energy in the B meson rest frame and fly apart at about 120 • angles. The corners regions correspond to the cases where one final state is soft and the others fly back-to-back with large energy about m B /2. At the edges of the Dalitz plot one invariant mass is small and the other two are large, which implies that two particles move collinearly and the third bachelor particle recoils back. The interactions between two collinear mesons leads eventually to the resonances. Compared with two other regions, the physics picture at the edges of the Dalitz plot is very similar to a two-body decay by viewing the two-meson pair as a whole, and we thus call it quasi-two-body decay. In the past twenty years, PQCD approach based on the k T factorization has been used to study the B meson two-body decays successfully, therefore it can be generalized for studying the quasi-two-body decays.
In the past few years, a large number of charmless quasi-two-body B/B s → V( f 0 (980) →)π + π − decays have been measured in the experiments [24], and some branching fractions or upper limits of them are summarized as follows (1) Except the decay B s → φ f 0 , f 0 → π + π − , other decays have not been studied theoretically in the literatures. Motivated by this, we shall study above decays in PQCD approach, so as to further check the reliability of PQCD in multi-body decays and present more predictions.For the sake of convenience f 0 (980) is abbreviated to f 0 in the following context unless special statement.

II. FRAMEWORK
In the framework of PQCD, the decay amplitude A of B → V( f 0 (980) →)π + π − decay can be decomposed as the convolution where x i are the momentum fractions of the light quarks, b i are the conjugate variables of the quarks' transverse momenta k iT . Φ B and Φ V are the wave functions of the B mesons and vector mesons, while the Φ ππ is the S -wave ππ-pair wave function. These wave functions are non-perturbative and universal. The exponential term is the so-called Sudakov form factor caused by the additional scale introduced by the intrinsic transverse momenta k T , which suppresses the soft dynamics effectively [25,26].
is the hard kernel, which can be calculated perturbatively. The parameter t is the largest scale in the hard kernel, which ensures the higher order corrections as small as possible.
In PQCD, the most important inputs are the initial and final mesons' wave functions. For the B meson and the light vector mesons, their wave functions have been studied extensively and the inner parameters have been fixed by the well measured two-body B meson decays [27,28], so we will not discuss them in this work. For the ππ-pair, its S -wave wave function can be written as [29][30][31][32][33] with z being the momentum fraction of the light quark in the ππ-pair. The parameter ξ is the momentum fraction of one π meson in the ππ-pair. The momentum of the ππ-pair P satisfies the condition P 2 = ω 2 , ω being the invariant mass of ππ-pair. n = (1, 0, 0) and v = (0, 1, 0) are the light-like vectors. For the explicit expressions of the light-core distributions φ (s,t) S , we adopt the form [34,35] with Gegenbauer moment a s = 0.3 ± 0.2 [32]. F S (ω) is the time-like form factor. In particular, for a narrow intermediate resonance, the time-like form factor F S (ω) can be well described by the relative Breit-Wigner lineshape [36]. However, due to the remarkable interference between two decays f 0 → ππ and f 0 → KK, the relative Breit-Wigner lineshape cannot work well for the time-like form factor of f 0 . In this case, the Flatté lineshape is proposed to describe that of f 0 [36,37], which is given as with The g ππ and g KK are the coupling constants corresponding to f 0 → ππ and f 0 → KK decays, respectively, whose values are taken as g ππ = (0.165 ± 0.018) GeV 2 and g KK /g ππ = 4.21 ± 0.33 [36]. In addition, the factor F KK = e −αq 2 is introduced to suppress the f 0 width above the KK threshold. The parameter α is taken 2.0 ± 1.0 GeV −2 , which does not affect the predictions remarkably [38]. It is noted that this lineshape has been also adopted extensively in analyzing data in the LHCb experiment [38]. Although the quark model has achieved great successes, the underlying structures of the scalar mesons are not well established so far. There are many scenarios for the classification of the scalar mesons. One scenario is the naive 2-quark model, and the light scalar mesons below or near 1 GeV are identified as the lowest lying states. Another consistent picture [39] provided by the data implies that light scalar mesons below or near 1 GeV can be described by the q 2q2 , while scalars above 1 GeV will form a conventional qq nonet with with some possible glue content [40][41][42]. This picture can be used to interpret the mass degeneracy of f 0 and a 0 (980), the reason why the widths of κ(800) and σ(600) is broader than those of a 0 (980) and f 0 , and the large couplings of f 0 and a 0 (980) to KK. However, in practice it is hard for us to make quantitative predictions on B decays based on the four-quark picture for light scalar mesons as it involves the unknown form factors and decay constants that are beyond the conventional quark model. Hence, we here only discuss the two-quark scenario for f 0 . Moreover, some experimental evidences indicate the existence of the non-strange and strange quark contents in f 0 , we therefore regard it as a mixture of ss and nn = (uū + dd)/ √ 2 where θ is the mixing angle. Recent studies [43][44][45] show that the mixing angle θ lies in the ranges of 25 • < θ < 40 • and 140 • < θ < 165 • , and studies based on the B decays favor the later range.
With the initial and final wave functions, we can calculate the whole amplitude of each decay mode in PQCD approach. In the leading order, the diagrams contributing to the decay B + → ρ + π + π − are shown in the Fig.1. The first two diagrams are the emission type diagrams with the first one emitting the ππ-pair and the second one with the vector meson emitted. The last two are the annihilation type diagrams. Because the decay amplitudes are very similar to those presented in the ref. [22], for the sake of simplicity, we shall not present them in this work. The other parameters used in the numerical calculations, such as the mass of the mesons, CKM matrix elements and the life times of B mesons, are taken from the Particle Data Group [24]. FIG. 1: Typical Feynman diagrams for the quasi-two-body decay B + → ρ + π + π − in PQCD, where the black squares stand for the weak vertices, and large (purple) spots on the quark lines denote possible attachments of hard gluons. The green ellipse represent ππ-pair and the red one is the light bachelor ρ + meson.

III. NUMERICAL RESULTS
As aforementioned, the mixing angle of f 0 have not yet been determined. At first, we set θ as a free parameter and plot the branching fractions of B u,d,s → V f 0 → Vπ + π − decays dependent on it in Fig. 2, where the green bands are the allowed regions in the experiments. Combining the experimental data of B 0 → ρ 0 ( f 0 →)π + π − and B 0 → K * (892) 0 ( f 0 →)π + π − decays, we get the range of the mixing angle    135 • ≤ θ ≤ 155 • , which is consistent with the results obtained from φ → f 0 γ and f 0 → γγ. In our previous work [46], the decays B → K * 0,2 (1430) f 0 /σ have been investigated, and we obtained θ ≈ 145 • after comparing with experimental result [47]. If the mixing angle θ = 145 • is adopted, the branching ratios of the B 0 → ρ 0 ( f 0 →)π + π − and B 0 → K * (892) 0 ( f 0 →)π + π − decays are given as which well match the experimental measurements: In view of this, we present all calculated results of the CP-averaged branching fractions and the local direct CP asymmetries of the concerned decay modes with θ = 145 • in Table. I. For comparison, the available experimental data are also listed. One can find that adopting the appropriate wave functions of initial and final states, our predictions are in good agreement with the current experimental data, although there are only upper limits for the B 0 → ω( f 0 →)π + π − and B + → ρ + ( f 0 →)π + π − decays. Finally, we plot all the branching fractions dependent on the mixing angle θ, which may shed light on the mixing angle by combining the ongoing experimental measurements. We acknowledge that there are many uncertainties in our calculations. In this work, we mainly evaluate three kinds of errors, namely the parameters in wave functions of the initial and final states, the higher order and power corrections, and the CKM matrix elements, respectively. In fact, the first kind of errors come from the uncertainties of nonperturbative parameters, such as the B in the S -wave LCDAs of the ππ-pair, et.al. We can find from the table that this kind of uncertainties are dominant. Fortunately, this kind of uncertainties could be reduced with the developments of the experiments or other nonperturbative theoretical approaches in future. The second kind of errors arise from the unknown higher order of α s and higher power corrections, which are reflected by varying the Λ QCD = 0.25±0.05 GeV and the factorization scale t from 0.8t to 1.2t, respectively. The last ones are caused by the uncertainties of the CKM matrix elements.
If the narrow-width approximation (NWA) holds in these decays, the branching fraction of the quasitwo-body B meson decay can be decomposed as with R represents a resonance. If two decays have a same resonance, we then define a ratio as Based on the predictions in Table. I, the ratio between the B 0 → ρ 0 f 0 and B 0 → ω f 0 is given as which is agreement with the results of QCDF [48]. Within the isospin relation we obtain the relation which can be used to predict the branching fractions of the corresponding quasi-two-body B → V( f 0 → )π 0 π 0 decays. Now we turn to discuss the local direct CP asymmetries of these decays. In the quark level, B → K * ( f 0 →)π + π − , B s → ρ 0 /ω( f 0 →)π + π − , and B s → φ( f 0 (980) →)π + π − are induced byb →sqq transition, while B → (ρ, ω)( f 0 →)π + π − and B s → K * ( f 0 (980) →)π + π − are controlled byb →dqq transition. From Table. I, it is found that the local CP asymmetries of decays B 0 → K * 0 ( f 0 →)π + π − and B s → ρ 0 /ω( f 0 → )π + π − are very small, and the reason is that the tree diagrams contributions are both color and CKM elements suppressed. However, for the decay B + → K * + ( f 0 →)π + π − , because the spectator u quark enters into not only K * + meson but also ππ-pair, the contributions from tree and penguin operators are comparable, leading to a large CP asymmetry. For the decays B 0 → ρ 0 /ω( f 0 →)π + π − , although the contributions from tree operators are color suppressed, the destructive interference between the diagrams with vector meson emitted and ones with ππ-pair emitted decreases the effects of penguin operators remarkably, therefore the CP asymmetries are as small as about 10%. For B + → ρ + ( f 0 →)π + π − decay, its amplitude is more complicated. In eq. (7), if the mixing angle θ of f 0 (980) is an obtuse angle, the sign of nn is negative. The spectator u-quark of B + → ρ + ( f 0 →)π + π − can enter into both f 0 and ππ-pair, so the negative sign leads to the cancellation between two tree operators contributions. With the sizable contributions of penguin operators, the CP asymmetry of this decay is as large as −55%. The decay of B s →K * 0 ( f 0 →)π + π − is very similar to B + → ρ + ( f 0 →)π + π − , but the spectator is a strange quark. When the spectator enters into the kaon, both the tree and penguin operators contribute, and the tree operators are color suppressed. However, when it enters into the ππ-pair, only penguin operators play roles. Due to large interference between two kinds of above contributions, the large CP asymmetry in B s →K * 0 ( f 0 →)π + π − is reasonable. On the experimental side, these CP asymmetries have not been measured, and we hope these predictions can be tested in future.

IV. SUMMARY
In this work, we investigated the quasi-two-body B/B s → V( f 0 (980) →)π + π − decays in PQCD approach, assuming that f 0 (980) is a mixture of nn = (uū + dd)/ √ 2 and ss with the mixing angle θ. Within the S -wave two-pion wave function, both the branching fractions and the located CP asymmetries have been calculated. When the mixing angle θ is around 145 • , the obtained branching fractions of the B 0 → ρ 0 ( f 0 (980) →)π + π − and B 0 → K * 0 ( f 0 (980) →)π + π − are in good agreement with the experimental data, and other results could be tested in the future experiments. In addition, the branching fractions of B/B s → V( f 0 →)π 0 π 0 could be predicted based on the isospin symmetry, which can be measured in the LHCb and Belle II experiments. We note that the calculated CP asymmetries of the B + → ρ + ( f 0 →)π + π − and B s →K * 0 ( f 0 →)π + π − decays are very large, which can be tested in the ongoing experiments. We acknowledge that there are many uncertainties in the calculation, and the dominant one is the two-meson wave function. Therefore, more precise multi-meson wave functions from non-perturbative approach are needed.